We can use an amazing fact about the normal distribution
to define multidimensional normal distributions
in an elegant way. To pick a d
al normal distribution
, just pick each orthogonal coordinate independent
ly according to a normal distribution (the coordinates may have different standard deviation
s, if you like; the claims made in the sequel
Obviously, this defines a distribution on Rd. What is less obvious is that this isn't dependent on the specific set of coordinates chosen! But it turns out that if you pick some other orthogonal basis, and take coordinates according to that, the projections of this distribution on each new coordinate are normally distributed, and independent!
This situation is truly impressive (for instance, if we used uniform distributions instead of normal ones, we'd still get a uniform distribution on some box in the first step; but the axes of the box would be parallel to the axes used, and any rotated set of axes would not have uniform or independent projections!